Abstract

We discuss a new method for simultaneously probing translational, rotational, and vibrational dynamics in dilute colloidal suspensions using digital holographic microscopy (DHM). We record digital holograms of clusters of 1-μm-diameter colloidal spheres interacting through short-range attractions, and we fit the holograms to an exact model of the scattering from multiple spheres. The model, based on the T-matrix formulation, accounts for multiple scattering and near-field coupling. We also explicitly account for the non-asymptotic radial decay of the scattered fields, allowing us to accurately fit holograms recorded with the focal plane located as little as 15 μm from the particle. Applying the fitting technique to a time-series of holograms of Brownian dimers allows simultaneous measurement of six dynamical modes — three translational, two rotational, and one vibrational — on timescales ranging from 10−3 to 1 s. We measure the translational and rotational diffusion constants to a precision of 0.6%, and we use the vibrational data to measure the interaction potential between the spheres to a precision of ∼50 nm in separation distance. Finally, we show that the fitting technique can be used to measure dynamics of clusters containing three or more spheres.

Figures (10)

Cartoon of experimental system and of the depletion interaction. Shaded area around each polystyrene sphere shows the excluded volume around each sphere, set by the radius of the depletant particles. The centers of the depletant particles cannot enter the shaded areas. We simultaneously measure the translation, rotation, and vibrational dynamics of pairs of particles interacting through a depletion attraction with several kBT well depth.

Simulated holograms, calculated using the algorithms described in Sections 4.2 and 4.3, for colloidal clusters made of 1-μm-diameter polystyrene spheres in water. Blue diagrams are renderings of the particles in real space; they are oriented so that the incident light propagates into the page. (a) Single sphere. (b) Dimer, spheres in contact. (c) Dimer, spheres separated by 200 nm. The magnitude of the separation is larger than typically observed experimentally and has been chosen for clarity. (d) Dimer, spheres in contact, rotated 30° into the page and 45° axially. (e) Trimer, parallel to the detector plane.

Comparison between simulated hologram calculated using T-matrix method and simulated hologram computed from Lorenz-Mie superposition, for a dimer composed of 1.57 μm spheres. As shown in the blue rendering, the upper particle is rotated 45° into the page. The rendering is oriented so that the incident light direction is into the page. (a) Hologram intensity along red dashed lines in (b) and (c). The hologram calculated from a T-matrix solution (blue) differs qualitatively from the hologram calculated by superposing the Lorenz-Mie solution for two spheres (green) due to near-field coupling. (b) Simulated hologram computed from T-matrix code. (c) Simulated hologram calculated by Lorenz-Mie superposition.

Hologram of a dimer composed of 1 micron spheres, held together by a depletion interaction. (a) Comparison of the recorded hologram (solid black lines) to the best fit, as calculated from the T-matrix scattering model (red symbols), along the three dashed lines indicated. (b) Recorded hologram. (c) Best fit. The blue diagram above the holograms shows a rendering of the particle positions from the fit. The upper sphere is rotated 34.9° into the page. The rendering is oriented so that the incident light direction is into the page.

Translations of dimer center-of-mass in the laboratory frame. Left: histograms of the distributions of cluster center-of-mass displacements in the x direction at 20, 100, and 500 ms. Solid lines are fits to a Gaussian. Right: Laboratory-frame mean-square displacements. Solid line represents linear (diffusive) behavior. Error bars are at most comparable in size to the plot symbols.

Measured translational mean-square displacements parallel and perpendicular to dimer axis. The solid lines are linear fits from which we determine D‖ and D⊥. Left: linear plot; our errorbars are shown by the dashed lines. Right: same data shown on a log-log plot.

Measured pair potential for a colloidal dimer. Only differences in U(r) are relevant; the actual values are arbitrary. The measured potential is qualitatively consistent with an attractive depletion force and an electrostatic repulsion. The bin width of the histogram of particle center-to-center separations, from which we determine the distribution of separations and the potential, is 11.7 nm.

Hologram of a trimer of 1.3 micron spheres. a) Comparison between the recorded hologram (solid black line) and the best fit, as calculated from the T-matrix scattering model (red symbols), along the three dashed lines indicated. b) Recorded hologram. c) Best fit. The blue diagram above the holograms shows a rendering of the particle positions from the fit. The leftmost sphere is rotated 38.3° into the page. The rendering is oriented so that the incident light direction is into the page.